What this calculator does
Slant height is the distance along the sloped outer surface of a cone or pyramid, from the base up to the apex. This is different from the vertical height, which runs straight up through the middle. Because the vertical height, the base measurement and the slant height form a right triangle, slant height is found directly from the other two using Pythagoras' theorem.
This calculator covers both a cone (from its base radius and vertical height) and a square pyramid (from its base side length and vertical height), since the same underlying triangle relationship applies to each shape with a slightly different base measurement.
The formula
For a cone, slant height = √(radius² + height²). For a square pyramid, slant height = √(height² + (half the base side)²), since the vertical height meets the base at its centre, half a base-side away from the midpoint of each side.
| Term | Meaning |
|---|---|
| Slant height | The straight-line distance from the base up to the apex, measured along the sloped surface rather than straight up. |
| Vertical height | The straight up-and-down height from the base plane to the apex, forming a right angle with the base. |
The inputs explained
| Field | What to enter |
|---|---|
| Shape | Choose whether the shape is a cone or a square pyramid; each uses a slightly different base measurement. |
| Base radius (cone) | The base radius, used only for a cone. |
| Base side length (pyramid) | The length of one side of the square base, used only for a square pyramid. |
| Vertical height | The vertical height from the base to the apex. |
When to use it
Estimating material for a cone or pyramid-shaped structure
The sloped surface area of a cone or pyramid (a marquee, a roof, a decorative object) is calculated from its slant height, not its vertical height, so finding the slant height first is often the needed step.
A geometry problem given height and base
A classic Pythagoras application: converting a known vertical height and base measurement into the slant height along the sloped face.
Checking a stated slant height
Recalculating slant height independently from the base and vertical height confirms whether a stated figure is consistent with those two measurements.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does slant height change with vertical height, for a cone with a 3-unit base radius?
The same base radius, with vertical height varied.
Questions
What's the difference between slant height and vertical height?
Vertical height goes straight up from the centre of the base to the apex. Slant height runs along the outside sloped surface from the edge of the base to the apex: it's always the longer of the two, since it's the hypotenuse of a right triangle that has vertical height as one side.
Does this work for a rectangular (non-square) pyramid base?
No, this covers a square base specifically, where every side is the same length and the slant height to the midpoint of each side is identical. A rectangular base has two different slant heights, one for each pair of sides.
How is this different from finding a cone's height from its slant height?
They're the reverse of each other. This calculator finds slant height starting from the vertical height and base; the height of a cone calculator starts from a known slant height and radius and finds the vertical height instead.
Can I use this for a triangular-based pyramid?
No, a pyramid with a triangular base doesn't have a single consistent "base side" the way a square pyramid does, so this calculator's pyramid mode is scoped to a square base specifically.
To go the other direction for a cone, see the height of a cone from slant height calculator. For a cone's full volume, see the cone volume calculator.